The constructivist or reform mathematics movement tends to set-up a distinction between the kind of mathematics that is often taught in school and the mathematics that is used in real-life. Three kinds of research are used to support this.
- Studies that show that children invent their own mathematical strategies if required to do so in order to earn a living. These include studies of Brazilian street vendors.
- Studies that show that adults who can answer an arithmetic question in a real-life context are less successful when answering similar questions in a formal test.
- Studies that show that children who are taught a standard mathematical procedure will apply it in a nonsensical way and/or generate bizarre answers.
Put together, this evidence is used to support a narrative that there is something artificial about school mathematics. Instead, we should teach students mathematics through involving them in authentic, real-life experiences of using maths. By developing their own strategies for solving such problems, they will have a better understanding of what they are doing and also be able to apply maths in a range of different contexts.
But it is worth analysing these ideas from a different perspective.
The Brazilian street vendors seem to show that it is possible to learn through discovery, although it seems likely that mathematical strategies are communicated between children or between adults and children, and so participants may simply be using procedures that someone else has shown them. It’s also worth noting that these students could have been at school until the second grade and still be included in the study.
A possibility of learning though discovery is not the same thing as discovery being the most effective way of learning. For instance, we know that many children will learn to read through a whole language approach. The problem with whole language is that fewer children will be successful this way and those that do learn may have later problems with spelling or pronouncing unfamiliar words.
The second set of studies relates to a failure of transfer. Transfer is notoriously difficult to achieve. Dan Willingham explains this as an inability of novices to tell the difference between the surface (i.e. unimportant) features of a question and the deep structure.
A fairly traditional way to teach for transfer is to start by teaching a concept in a particular context or in the abstract and then gradually work outwards to more diverse contexts. I also think activities that compare and contrast approaches can help students apprehend the deep structure, something that I wrote a little about here. It is interesting that contrasting cases appear a great deal in studies on ‘productive failure’ and ‘preparation for future learning’ literature, although not always in the explicit instruction control conditions that are used.
It seems likely that students who learn maths through a particular real-life context may have their learning stuck to that context and may struggle to transfer their learning just as much as those students who learnt ‘school mathematics’. Moreover, the very artificiality of school maths is often about stripping away surface features in order to make the deep structure more visible – this is the point of maths being abstract. Students who have learnt school mathematics may well struggle to apply it to a novel measurement problem but those who learnt the same mathematics through solving money problems may struggle even more.
Finally, what of the students who, when taught standard mathematical procedures, use them to give bizarre answers or answers that don’t fit the context. That was the result of the Kamii and Dominick study that I linked to above. They found that students who were taught to use the standard procedure often produced answers that were a long way from the real answer whereas even when students who invented their own strategy were wrong, their wrong answers were a good estimate for the actual answer. It was not a true experiment and, when a similar study was conducted by Stephen Norton in Queensland, a very different result was obtained.
There were two key differences between Kamii and Dominick and Norton: Norton classified children based upon their answers – had they used the standard algorithm or not? – whereas Kamii and Dominick tracked students who had been taught in particular ways. Secondly, the calculations that students did in the Norton paper where generally more complex, involving larger numbers. Norton found that students who used the standard approach faired better.
Bearing this in mind, it is worth reviewing an expert panel report produced in 2004 for the Ontario government in Canada. In states:
“It is known that, initially, most children come to school as enthusiastic, curious thinkers, whose natural inclination is to try to make mathematical sense of the world around them. This natural curiosity can be nurtured in a problem-solving approach that begins with, and fosters, student own ideas and methods. For example, Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) found that two-thirds of Kindergarten and Grade 1 students in mathematics programs focused on problem solving were able to solve the following problem: If a class of 19 children is going to the zoo and each car can take 5 children, how many cars are needed? When asked whether all the cars were full, they said: “No, there is an extra seat in one car” or “Yes, because I’m going too!” They were making sense of the question. Contrast these findings with test results of Grade 8 students in non-problem-solving programs who were asked the same type of question, but with larger numbers: An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed? Two thirds of the 45 000 students tested performed the long division correctly. However, some wrote that “31, remainder 12” buses were needed, or just 31 – lopping off the remainder. Only one-quarter of the total group gave the correct answer of 32 buses (O’Brien, 1999). For those students, learning “school mathematics” (Fosnot & Dolk, 2001b) meant carrying out procedures without making sense of what they were doing. Is there evidence that Ontario students stop making sense of what they are doing in mathematics as they progress through school?
The clear implication is that learning standard approaches harms students’ ability to make sense of maths problems: algorithms are harmful. Yet, when we look at the examples used to make this case, two things are changing at once; the familiarity with standard procedures and the complexity of the problems (setting aside the fact that we are talking about very different cohorts of students). Would students unfamiliar with the standard approaches have fared better on the soldiers problem? I think the Norton research suggests that they might not.
I think I know what is going on here. For most students, carrying out a long division problem is sufficient to fill-up the working memory, so much so that there is little capacity left over to attend to other aspects of the question. This is why students make supposedly ‘silly’ mistakes in class. The solution to this is to practice long division to the point that it is more automatic and requires less conscious attention. This way, students can focus more on the context of the question and the reasonableness of their answers.
This is quite the opposite conclusion to the constructivist prescription for more real-life, experiential learning.